Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 006, 60 pages      math-ph/0601037      http://dx.doi.org/10.3842/SIGMA.2006.006

Orbit Functions

Anatoliy Klimyk a and Jiri Patera b
a) Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv, 03143 Ukraine
b) Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received January 04, 2006; Published online January 19, 2006

Abstract
In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space En are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space En. Orbit functions are solutions of the corresponding Laplace equation in En, satisfying the Neumann condition on the boundary of F. Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.

Key words: orbit functions; Coxeter-Dynkin diagram; Weyl group; orbits; products of orbits; orbit function transform; finite orbit function transform; Neumann boundary problem; symmetric polynomials.

pdf (599 kb)   ps (375 kb)   tex (56 kb)

References

  1. Patera J., Orbit functions of compact semisimple Lie groups as special functions, in Proceedinds of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv, 2004, V.50, Part 3, 1152-1160.
  2. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions, Vols. 1-3, Dordrecht, Kluwer, 1991-1993.
  3. Miller W., Lie theory and special functions, New York, Academic Press, 1968.
  4. Vilenkin N.Ja., Special functions and the theory of group representations, Providence RI, Amer. Math. Soc., 1968.
  5. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford, Oxford Univ. Press, 1995.
  6. Macdonald I.G., A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20, 1988, Séminaire Lotharingien, 131-171.
  7. Macdonald I.G., Orthogonal polynomials associated with root systems, Séminaire Lotharingien de Combinatoire, Actes B45a, Strasbourg, 2000.
  8. Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions: recent advances, Dordrecht, Kluwer, 1995.
  9. Moody R.V., Patera J., Computation of character decompositions of class functions on compact semisimple Lie groups, Math. Comp., 1987, V.48, 799-827.
  10. Moody R.V., Patera J., Elements of finite order in Lie groups and their applications, XIII Int. Colloq. on Group Theoretical Methods in Physics, Editor W. Zachary, Singapore, World Scientific Publishers, 1984, 308-318.
  11. McKay W.G., Moody R.V., Patera J., Tables of E8 characters and decomposition of plethysms, in Lie Algebras and Related Topics, Editors D.J. Britten, F.W. Lemire and R.V. Moody, Providence R.I., Amer. Math. Society, 1985, 227-264.
  12. McKay W.G., Moody R.V., Patera J., Decomposition of tensor products of E8 representations, Algebras Groups Geom., 1986, V.3, 286-328.
  13. Patera J., Sharp R.T., Branching rules for representations of simple Lie algebras through Weyl group orbit reduction, J. Phys. A: Math. Gen., 1989, V.22, 2329-2340.
  14. Grimm S., Patera J., Decomposition of tensor products of the fundamental representations of E8, CRM Proc. Lecture Notes, 1997, V.11, 329-355.
  15. Atoyan A., Patera J., Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization, J. Math. Phys., 2004, V.45, 2468-2491, math-ph/0309039.
  16. Rao K.R., Yip P., Discrete cosine transform - algorithms, advantages, applications, New York, Academic Press, 1990.
  17. Kane R., Reflection groups and invariants, New York, Springer, 2002.
  18. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge, Cambridge Univ. Press, 1990.
  19. Humphreys J.E., Introduction to Lie algebras and representation theory, New York, Springer, 1972.
  20. Pinsky M.A., The eigenvalues of an equilateral triangle, SIAM J. Math. Anal., 1980, V. 11, 819-827.
  21. Patera J., Algebraic solution of the Neumann boundary problems on fundamental regions of a compact semisimple Lie group, CRM Preprint, Montreal, 2003.
  22. Patera J., Compact simple Lie groups and their C-, S-, and E-transforms, SIGMA, 2005, V.1, paper 025, 6 pages, math-ph/0512029.
  23. Bremner M.R., Moody R.V., Patera J., Tables of dominant weight multiplicities for representations of simple Lie algebras, New York, Marcel Dekker, 1985.
  24. McKay W.G., Patera J., Rand D.W., Tables of representations of simple Lie algebras, Montreal, CRM, 1990.
  25. Champagne B., Kjiri M., Patera J., Sharp R.T., Description of reflection generated polytopes using decorated Coxeter diagrams, Can. J. Phys., 1995, V.73, 566-584.
  26. Moody R.V., Patera J., Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams, J. Phys. A: Math. Gen., 1992, V.25, 5089-5134.
  27. McKay W.G., Patera J., Sannikoff D., The computation of branching rules for representations of semisimple Lie algebras, in Computers in Nonassociative Rings and Algebras, Editors R.E. Beck and B. Kolman, New York, Academic Press, 1977, 235-278.
  28. Gingras F., Patera J., Sharp R.T., Orbit-orbit branching rules between simple low-rank algebras and equal-rank subalgebras, J. Math. Phys., 1992, V.33, 1618-1626.
  29. Patera J., Sharp R.T., Branching rules for representations of simple Lie algebras through Weyl group orbits reduction, J. Phys. A: Math. Gen., 1989, V.22, 2329-2340.
  30. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to Lie groups SU(2)×SU(2) and O(5), J. Math. Phys., 2005, V.46, 053514, 25 pages.
  31. Patera J., Zaratsyan A., Discrete and continuous cosine transform generalized to Lie groups SU(2) and G2, J. Math. Phys., 2005, V.46, 113506, 17 pages.
  32. Lemire F.W., Patera J., Congruence number, a generalisation of SU(3) triality, J. Math. Phys., 1980, V.21, 2026-2027.
  33. Zhelobenko D.P., Compact Lie groups and their representations, Moscow, Nauka, 1970.
  34. McKay W.G., Patera J., Tables of dimensions, indices and branching rules for representations of simple Lie algebras, New York, Marcel Dekker, 1981.
  35. Strang G., The discrete cosine transform, SIAM Review, 1999, V.41, 135-147.
  36. Kac V., Automorphisms of finite order of semisimple Lie algebras, J. Funct. Anal. Appl., 1969, V.3, 252-255.
  37. Moody R.V., Patera J., Characters of elements of finite order in simple Lie groups, SIAM J. Algebraic Discrete Methods, 1984, V.5, 359-383.
  38. McKay W.G., Moody R.V., Patera J., Pianzola A., The 785 conjugacy classes of rational elements of finite order in E8, Contemp. Math., 1990, V.110, 79-123.
  39. Heckman G.J., Opdam E.M., Root systems and hypergeometric functions. I, Compos. Math., 1987, V.64, 329-352.
  40. Heckman G.J., Root systems and hypergeometric functions. II, Compos. Math., 1987, V.64, 353-373.
  41. Gasper G., Rahman M., Basic hypergeometric functions, Cambridge, Cambridge Univ. Press, 1990.

Previous article   Next article   Contents of Volume 2 (2006)